Higher-Order MDS Codes

An improved Singleton-type upper bound is presented for the list decoding radius of linear codes, in terms of the code parameters
$[n,k,d]$
and the list size
$L$
.
$L$
-MDS codes are then defined as codes that attain this bound (under a slightly stronger notion of list decodability), with 1-MDS codes corresponding to ordinary linear MDS codes. Several properties of such codes are presented; in particular, it is shown that the 2-MDS property is preserved under duality. Finally, explicit constructions for 2-MDS codes are presented through generalized Reed–Solomon (GRS) codes.